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Proof that Casimir force does not originate from vacuum energy Hrvoje Nikolić arXiv:1605.04143v1 [hep-th] 13 May 2016 Theoretical Physics Division, Rudjer Bošković Institute, P.O.B. 180, HR-10002 Zagreb, Croatia.∗ (Dated: May 16, 2016) We present a simple general proof that Casimir force cannot originate from the vacuum energy of electromagnetic (EM) field. The full QED Hamiltonian consists of 3 terms: the pure electromagnetic term Hem , the pure matter term Hmatt and the interaction term Hint . The Hem -term commutes with all matter fields because it does not have any explicit dependence on matter fields. As a consequence, Hem cannot generate any forces on matter. Since it is precisely this term that generates the vacuum energy of EM field, it follows that the vacuum energy does not generate the forces. The erroneous statements in the literature that vacuum energy generates Casimir force can be boiled down to the fact that Hem attains an implicit dependence on matter fields by the use of the equations of motion and the erroneous treatment of the implicit dependence as if it was explicit. The true origin of the Casimir force is van der Waals force generated by Hint . PACS numbers: 11.10.Ef, 03.70.+k, 42.50.Lc Introduction.—The Casimir force [1] is widely viewed as a force that originates from the vacuum energy, which is a view especially popular in the community of highenergy physicists [2–7]. Another view, more popular in the condensed-matter community, is that Casimir force has the same physical origin as van der Waals force [8–14], which does not depend on energy of the vacuum. From a practical perspective, the two points of view appear as two complementary approaches, each with its advantages and disadvantages. From a fundamental perspective, however, one may be interested to know which of the two approaches is more fundamental. After all, the conceptual picture of the world in which the vacuum energy has a direct physical role is very different from the picture in which it does not. From such a fundamental perspective, Jaffe argued [15] that the physically correct approach is the one based on van der Waals force, while the approach based on vacuum energy is merely a heuristic shortcut valid only as an approximation in the limit of infinite fine structure constant. Nevertheless, it seems that a general consensus is still absent. The question of relevance of the vacuum energy for Casimir force is still a source of controversy. In this paper we present a theoretical way to resolve the controversy. In short, similarly to Jaffe [15], we find that the approach based on vacuum energy is unjustified from a fundamental theoretical perspective, leaving only the non-vacuum van der Waals-like approaches as physically viable. However, to arrive at that conclusion, we use an approach very different from the approach used by Jaffe. Our approach is rather mathematical in spirit, because our central idea is to carefully distinguish explicit dependence from implicit dependence in canonical equations of motion for classical and quantum physics. In this way our approach is more abstract and more general than the approach by Jaffe, but still sufficiently simple to be ∗ Electronic address: [email protected] accessible to a wide readership of theoretical physicists. Heuristic idea.—Let us start with a brief overview of the standard calculation [2] of Casimir force from vacuum energy. The energy of electromagnetic (EM) field is Z E 2 + B2 . (1) Hem = d3 x 2 In general, the fields E and B have Fourier transforms with contributions from all possible wave vectors k. However, in the absence of electric currents in a conductor, the use of Maxwell equations implies that the EM field must vanish at conducting plates. Consequently, if there are two conducting plates separated by a distance y, then the only wave vectors in the y-direction that contribute to E and B are those which satisfy ky = nπ/y (for n = 1, 2, 3, . . .). In this way E and B attain a dependence on y, which we write as E → Ẽ(y), B → B̃(y). Consequently we have Hem → H̃em (y), which leads to a y-dependent vacuum energy Ẽvac (y) = h0|H̃em (y)|0i. (2) The general principles of classical mechanics then suggest that there should be the force between the plates given by F̃ (y) = − ∂ Ẽvac (y) . ∂y (3) A detailed calculation [2] shows that Ẽvac (y) = Ẽfin (y) + E0 , where Ẽfin (y) is finite and E0 is an infinite constant that does not depend on y. In this way (3) gives a finite result that turns out to agree with measurements [16]. The central idea of this paper is to question the validity of Eq. (3), the equation which in the existing literature is usually taken for granted without further scrutiny. If (3) is valid, then y must be a dynamical variable with a kinetic part in the full Hamiltonian. Treating plates as classical non-relativistic objects, the minimal Hamiltonian that leads to (3) is H̃ = p2y + Ẽvac (y), 2m (4) 2 where m = m1 m2 /(m1 + m2 ) is the reduced mass of the two plates with masses m1 and m2 . From the point of view of general principles of classical mechanics [17, 18], however, there is something suspicious about (3) and (4). The term Ẽvac (y) in (4) creates the force (3) on the dynamical variable y(t) owing to the fact that Ẽvac (y) depends on y. On the other hand, Ẽvac (y) originates from (1) which does not depend on y, so Hem cannot generate any force on y because ∂Hem /∂y = 0. This looks like a paradox; how can it be that Hem both depends and does not depend on y? The answer, of course, is that Hem does not have any explicit dependence on y. Yet H̃em (y) has an implicit dependence on y, i.e. the dependence which originates from using solutions of the equations of motion. Now what kind of y-dependence is responsible for forces, explicit or implicit? The general principles of classical mechanics [17, 18] tell us that it is only explicit dependence that counts! Consequently, (3) is not a legitimate formula to calculate the force. In this way we see that our critique of Eq. (3) has the origin in general principles of classical mechanics. Nevertheless, in this paper we shall perform a full quantum analysis and we shall see that a similar critique of (3) works also in quantum physics. The essence of the error committed in (3) is treating the implicit dependence as if it was explicit, which is equally wrong in both classical and quantum theory. The main proof.—The full action of quantum electrodynamics (QED) can be written as I = Iem (A) + Imatt (φ) + Iint (A, φ), (5) where πA and πφ are the canonical momenta and each of the 3 terms is generated by the corresponding term in (5). The time evolution of quantum variables is governed by the Hamiltonian. We use the Heisenberg picture, so matter variables φ and πφ obey the Heisenberg equations of motion φ̇ = i[H, φ], π̇φ = i[H, πφ ]. (10) In particular, the quantity i[H, πφ ] on the right-hand side of the second equation in (10) describes the quantum force on the matter. We stress that (10) is exact, so all quantum forces on matter that can be derived from QED are described by (10). In particular, (10) is a nonperturbative result, so in principle it contains effects of all higher order quantum loop diagrams. Of course, the contributions of loop diagrams are not explicit because we work in the non-perturbative Heisenberg picture. Loop diagrams are a perturbative concept, so they could be seen explicitly if we worked in the Dirac interaction picture. Now the crucial observation is the fact that Hem (A, πA ) does not have any explicit dependence on φ and πφ . Consequently [Hem , φ] = 0, [Hem , πφ ] = 0, (11) so (10) reduces to φ̇ = i[Hint , φ] + i[Hmatt , φ], π̇φ = i[Hint , πφ ] + i[Hmatt , πφ ]. (12) where A(x) = {Aµ (x)} is the EM field and φ(x) denotes all matter fields. (The same notation X(Y ) is used for both functions such as φ(x) and functionals such as Iem (A). The meaning of this uniform notation in each particular case should be clear from the context.) Explicitly Z 1 Iem (A) = − d4 x F µν Fµν , (6) 4 Thus we see that all quantum forces on matter are generated by Hint and Hmatt . In other words, we have proven that Hem does not have any contribution to the quantum force on matter. So far our analysis was very general and we said nothing specific about the vacuum energy of EM field. To see the role of vacuum energy consider a state of the form Z where |0A i is the EM vacuum and |ψφ i is an arbitrary matter state. In the EM vacuum we have h0A |Aµ |0A i = 0, so using the fact that Hint is linear in Aµ due to (7), we have Iint (A, φ) = − 4 µ d x Aµ j (φ), (7) where Fµν = ∂µ Aν − ∂ν Aµ and j µ (φ) is the charge current. The explicit expressions for Imatt (φ) and j µ (φ) will not be needed. The action (5) can be written in terms of a Lagrangian L as Z I(A, φ) = dt L(A, Ȧ, φ, φ̇), (8) where the dot denotes the time derivative. Then by standard canonical methods [19] one can transform Lagrangian L into the Hamiltonian H = Hem (A, πA ) + Hmatt (φ, πφ ) + Hint (A, φ, πφ ), (9) |Ωi = |0A i|ψφ i, hΩ|Hint |Ωi = 0. (13) (14) On the other hand Hem is quadratic in Aµ due to (6), so all EM vacuum energy comes from the term Evac = hΩ|Hem |Ωi = h0A |Hem |0A i. (15) Of course, the term hΩ|Hmatt |Ωi = hψφ |Hmatt |ψφ i also contributes to the total energy in the EM vacuum, but this is clearly the matter energy determined by the matter state |ψφ i, so it cannot be considered as a contribution to the EM vacuum energy. 3 To conclude, from (12) we see that Hem does not contribute to quantum forces on matter, and from (15) we see that only Hem contributes to the EM vacuum energy. This proves our main result that EM vacuum energy does not contribute to quantum forces on matter. In principle our paper could stop here. Nevertheless it may be illuminating to consider some additional issues. If, as we just proved, vacuum energy does not create forces on matter, then what is wrong with Eq. (3)? In the rest of the paper we provide a deeper understanding of the error committed in (3). An illegitimate use of the equations of motion.—In the Lorenz gauge ∂µ Aµ = 0 the equations of motion for EM field are ✷Aµ = −j µ (φ), (16) To get the Casimir force one can model matter field φ(x, t) with a single degree of freedom y(t) representing the distance between the Casimir plates, in which case (23) reduces to ṗy = − µ A = à (φ), hΩ|ṗy |Ωi = −hΩ| õ (φ) ≡ −✷−1 j µ (φ). (18) In this way the use of equations of motion allows one to express the EM field as a functional of matter fields. Thus, even though Aµ does not have an explicit dependence on φ, it depends on φ implicitly due to (17). Inserting (17) into the expression for Hem (A, πA ) one gets the quantity H̃em (φ) = Hem (Ã(φ), π̃A (φ)). (19) In this way one gets non-vanishing commutators [H̃em , φ] 6= 0, [H̃em , πφ ] 6= 0, (20) despite the fact that the commutators (11) vanish. So far we did nothing wrong. But now consider the following step. Guided by the correct Heisenberg equations of motion (10), one may be tempted to write ? π̇φ = i[H̃em, πφ ]. (21) For the moment let us pretend that (21) was legitimate, to see where that would lead us. If that was legitimate, then the quantity i[H̃em , πφ ] would represent a nonvanishing force generated by H̃em . In the EM vacuum state (13) we would have hΩ|π̇φ |Ωi = ihΩ|[H̃em , πφ ]|Ωi. (22) so we would get a force that originates from the EM vacuum. When φ is a bosonic field, the momentum operator can be represented as πφ = −iδ/δφ, so the second equation in (21) can be written as π̇φ = − δ H̃em . δφ (23) ∂ H̃em |Ωi. ∂y (25) The distance between the plates y(t) is a macroscopic observable, so it can be approximated by a classical variable yc (t). As a consequence, (25) can be approximated by ṗy,c = − (17) where (24) Hence in the EM vacuum we have with ✷ ≡ ∂ ν ∂ν . This equation of motion can be solved as µ ∂ H̃em . ∂y ∂hΩ|H̃em |Ωi . ∂yc (26) This is equivalent to Eq. (3), here obtained from a more general equation of motion (21). Finally, from a formal point of view it is interesting to see what happens if all microscopic variables are considered in the classical limit. In that limit all the commutators turn into Poisson brackets according to the rule [A, B] → i{Ac , Bc }, so the second equation in (21) turns into π̇φ,c = {πφ,c , H̃em,c } = − δ H̃em,c . δφc (27) This classical equation has the same form as the quantum equation (23). Nevertheless, (21) was not legitimate. Consequently, all equations (22)-(27) above were illegitimate. When calculating commutators in canonical equations of motion in the Heisenberg picture, one must use H, not H̃. In other words, to calculate the commutators in the Heisenberg equations of motion, what counts is the explicit dependence [20–23], not the implicit dependence obtained by solving the equations of motion. In general, the use of equations of motion in such a way leads to wrong results. Next we demonstrate this fact explicitly, by solving a simple toy model. A toy model.—To see what goes wrong when equations of motion are used in a way described above, let us study a concrete example. Instead of field Aµ (x, t) with an infinite number of degrees of freedom, let us consider a single degree of freedom A(t). Similarly, instead of matter field φ(x, t), let us consider another single degree of freedom y(t). Let the dynamics of these two degrees of freedom be described by the Hamiltonian H = HA + Hy + Hint , (28) where HA = p2y p2A , Hy = , 2 2 Hint = −γ 2 Ay, (29) 4 and γ 2 is a coupling constant. A straightforward calculation shows that the resulting equations of motion are Ä = FA , ÿ = Fy , (30) where the forces are given by FA = − ∂H ∂H = γ 2 y, Fy = − = γ 2 A. ∂A ∂y (31) Let us study one particular solution of (30) A(t) = e−γt , y(t) = e−γt . (32) Inserting the solution into the second equation in (31) we obtain Fy = γ 2 y, (33) which is the correct result. Now using pA = Ȧ, the first equation in (29) can be written as HA = Ȧ2 . 2 (34) Using the solution (32), we can write Ȧ = ẏ = −γy. Inserting this into (34), we get H̃A (y) = γ 2y2 . 2 Our analysis based on Heisenberg picture, which is very suitable for proving general results, cannot easily answer such a more specific question. This question can be answered by a very different method [15], based on perturbative expansion in Feynman diagrams. The force calculated in terms of vacuum energy turns out not to be an exact result, but an approximation corresponding to the limit in which fine structure constant goes to infinity [15]. In this sense Casimir force, like any other force in quantum field theory, originates from the interaction term Hint . More precisely, as stressed in [15], the physical origin of Casimir force lies in the van der Waals forces between the conducting plates. Similarly to the vacuum energy, the van der Waals forces also originate from quantum fluctuations. However, the important difference lies in the fact that van der Waals forces do not originate from vacuum fluctuations. The van der Waals force originates from matter fluctuations of charge density [26, 27], but it does not depend on the operator ordering of charge current operator R j µ (φ) in the interaction term Lint = − d3 x Aµ j µ (φ). The force persists even if j µ (φ) is normal ordered so that h0φ |j µ (φ)|0φ i = 0. As a side remark let us also note that the charge current must be normal ordered for physical reasons. Without normal ordering one would have h0φ |j µ (φ)|0φ i 6= 0 [28], so the Maxwell equation (35) ∂µ F µν = −j ν (37) From this one may be tempted to calculate the force as F̃y = − ∂ H̃A = −γ 2 y. ∂y (36) However, comparing (36) with (33) one can see very explicitly that F˜y 6= Fy . This explicitly demonstrates that (36) is not a legitimate way to calculate the force. In the simple case above the forces (36) and (33) turn out to have the same absolute value and the opposite sign (the reader is encouraged to check all the signs by himself/herself), but this is not a general rule. The only rule is that F˜y and Fy are, in general, different. The possibility that F˜y and Fy turn out to give the same result in some case is not excluded, but this is an exception and we are not aware of any simple example of such a case. Discussion and conclusion.—We have shown that calculating Casimir force by (3) is not legitimate. Yet, it is known that such a calculation leads to a result which is in agreement with experiments, as well as with results obtained by other methods that do not involve vacuum energy. The methods with and without vacuum energy give the same results not only for perfect conductors, but even for realistic materials (compare e.g. [24] with [25]). How can it be that an illegitimate method leads to a correct result? would imply existence of EM fields even in the vacuum, in a clear contradiction with observations. We also note that taking normal ordering of j µ in the Maxwell equation is analogous to taking normal ordering in the energymomentum tensor T µν in the Einstein equation, the latter being closely related to the cosmological constant problem in gravitational physics [29–32]. To conclude, in this paper we have shown that the standard calculation of Casimir force from the vacuum energy through Eq. (3) is illegitimate. Essentially, this is because the vacuum energy has only implicit dependence on the distance y between the plates, namely the dependence that originates from the solution of the equations of motion, while the legitimate application of (3) would require an explicit dependence on y, which is absent. Therefore, at the fundamental level, the Casimir force does not originate from the vacuum energy. This is in full accordance with the result by Jaffe [15], according to which the true physical origin of the Casimir force lies in the non-vacuum van der Waals forces between the material plates. Acknowledgments.—The author is grateful to M.S. Tomaš for discussions. This work was supported by the Ministry of Science of the Republic of Croatia. 5 [1] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetensch. 51, 793 (1948). [2] C. Itzykson, J-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980). [3] N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space (Cambridge Press, New York, 1982). [4] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 2002). [5] V. Mukhanov, S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, Cambridge, 2007). [6] A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, Princeton, 2010). [7] T. 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